Area moment of inertia
Ixx = ∫ y² dA
Material far from the bending axis counts with its distance squared. A standing I-beam has a large Ixx, so it is stiff vertically, and a small Iyy, so it bends easily sideways. Same beam, same material, different axis.
Bending stiffness is E·I. A steel and an aluminium beam of the same shape have the same I but a different EI.
Not the same as mass moment of inertia (kg·m²), which governs how hard a body is to spin. Try it with a plastic ruler: bend it across its flat face, then across its edge.
Principal stress
σ1,2 = (σx + σy)/2 ± √[ ((σx − σy)/2)² + τxy² ]
The six stress components at a point depend on the axes you choose. Rotate the axes until every shear stress is zero: the normal stresses left are the principal stresses. σ1 is the largest tension on any plane through the point, and σ3 the largest compression.
There are always three, on mutually perpendicular planes, because the stress tensor is a symmetric 3×3 matrix. The principal stresses are its eigenvalues and the principal directions its eigenvectors.
Von Mises or principal stress?
Von Mises folds all six components into one positive number to compare with the yield strength from a tensile test. It suits ductile metals: steel and aluminium.
It has no sign and no direction, so it cannot tell tension from compression, and a brittle crack can hide on a von Mises fringe. For glass, castings, lamp lenses and plastic clips, judge with σ1. Cracks open perpendicular to its direction.
Once most of a part has yielded in an impact, effective plastic strain says more than either.
Energy absorption
E = ∫ F ds ≈ Favg × stroke
Energy is the area under the force–displacement curve, not its peak. A structure that is too stiff reaches a high force with little crush and passes the load downstream. One that is too soft bottoms out before it has absorbed enough.
The aim is a flat, sustained crush force over a long stroke. At a comparable force level, the longer available stroke wins, which is why standoff distance is often the best way to rank two designs before running any FEA.
Compare designs of different mass with specific energy absorption, SEA = E / mass. Judge how flat the curve is with crush force efficiency, CFE = mean force / peak force; closer to 1 is better. A higher I tunes how a section folds; it does not raise its capacity.
Stress is not force
Stress is force per area. A thick bracket can carry a large force at low stress, and a thin flange can show high stress while carrying little.
Principal-stress vectors show where a load path runs. A section force (*DATABASE_CROSS_SECTION, written to secforc) tells you how much force goes through it. It is the free-body-diagram cut from a mechanics course, done inside the model.
A useful check: put a cut on each branch of the path and confirm the branch forces add up to the applied load.
Load paths as stress trajectories
Join the principal directions point to point and you get stress trajectories. Those curves are the load path. In a simply supported beam with a centre load, compression arches from the load to the supports while tension hangs underneath like a cable, and the two families always cross at 90°. The part behaves like a hidden truss.
Adding material along a trajectory stiffens the path. A bead or notch across a compression trajectory makes the member buckle early instead of passing the load through.